I was referencing an image in the same folder as the HTML file, so that wasn't the problem.

Turns out, you have to copy the image into the Mobipocket subfolder that was created after you import the HTML file. It's a tricky hidden step that I just happened to stumble across some documentation online. I would have thought that Mobipocket would have referenced the linked files when it imported the HTML file, but no. You have to move them manually between importing and building your .prc file.